Question

Explain: Can two acute angles be supplementary?

Answer 1

Hint: Assume any two acute angles as angle A and angle B. Write the range of these acute angles mathematically and add the two sets of ranges to determine if two acute angles can be supplementary or not.

Complete step-by-step answer:
First let us see what the supplement of an angle means. Then we will understand the term acute angles.

Two angles are supplementary if their measures add up to 180 degrees or $\pi $ radians. Due to the fact that angles on a straight line sum 180 degrees it can be said that angles on straight lines are supplementary. For example: two angles measuring 120 degrees and 60 degrees are a supplement of each other.

Now, let us know about acute angles. An acute angle is a type of angle whose measure is less than 90 degrees. For example: 30 degrees or 60 degrees are some examples of acute angle. A triangle formed by all angles measuring less than 90 degrees is called acute angle triangle.
Now, let us come to the question. We have to determine whether two acute angles can be supplementary or not.

Assume that there are two acute angles namely, $\angle A\text{ and }\angle B$. We know that both angle are less than ${{90}^{\circ }}$, so if the sum of these angles is taken then it will be less than the sum of two ${{90}^{\circ }}$ angles, that is ${{180}^{\circ }}$. Hence, it is proved that two acute angles cannot be supplementary.

Note: As you can see the approach we have used to solve this question. We have considered the range of two angles and then taken their sum to find its range. We saw that the range of the sum of these angles is less than 180 degrees and therefore, two acute angles cannot be supplementary.